Lorenz attractor

Points on the Lorenz attractor projected onto the x, z plane

Edward Lorenz was using a primitive computer (it was 1963) to numerically integrate an apparently simple set of coupled differential equations. The computer worked to 6 decimal places and printed out each line to 3 places. Restarting a run, he noticed that the trace started looking similar but became slowly different to a previous run where the computer had not been restarted. Lorenz investigated and realised that the rounding error between restarting with the printed 3 place values and the continuous run 6 place values was sufficient to change the future unfolding of the system completely.

This sensitive dependence on initial conditions is the essence of mathematical chaos – Lorenz later coined the phrase butterfly effect to describe the sensitive dependence on initial conditions that some systems show. The punchline is that you can’t make predictions about the long term future of systems displaying the butterfly effect – the slightest measurement error on your input will lead to totally different values at some definite time in the future, although the system will remain within certain bounds.

Chipmunk basic code for lorenz attractor using simple euler integration

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